Dirichlet problem

1. The problem statement, all variables and given/known data

Find function ##u(x,y)## that is harmonic on the upper half-plane ##0<Im(z)##. Note that

##u(x,0)=0##, ##x<0##

##u(x,0)=-1##, ##0<x<1## and

##u(x,0)=1##, ##x>1##.





2. Relevant equations



##u(z)=\frac{1}{2\pi i}\int _{I}\gamma '(s)u(\gamma (s))\frac{\alpha '(\gamma (s))}{\alpha (\gamma (s))}Re(\frac{\alpha (\gamma (s))+\alpha (z)}{\alpha (\gamma (s))-\alpha (z)})ds##



Where ##\alpha ## is holomorphic function that maps ##Im(z)>0## into open unit disk, and ##\gamma (s) ## parameterization of the edge.



3. The attempt at a solution



For this problem ##\alpha (z)=\frac{1-iz}{1+iz}## and it is also obvious that I will have to make at least 3 integrals.



Let's firstly take a look at ##u(x,0)=0##, ##x<0##:



This tells me that ##u(z)=\frac{1}{2\pi i}\int _{I}\gamma '(s)u(\gamma (s))\frac{\alpha '(\gamma (s))}{\alpha (\gamma (s))}Re(\frac{\alpha (\gamma (s))+\alpha (z)}{\alpha (\gamma (s))-\alpha (z)})ds## will be equal to ##0## for all ##x<0##.



But the question here is: What about ##y## ? None of the conditions tell anything about ##y##. What do I do here?



The same question is for ##u(x,0)=-1##, ##0<x<1##:



Here the integral for ##x## goes from ##0## to ##1##. But again, ##y## can be anything from ##0## to ##\infty ##. ???



I guess my question here is: What should I do with ##y## to get ##u(z)##?